So, I tried factory resetting the device from fastboot. This does allow the device to get past the bouncing circles screen, but it isn't a permanent fix. When I first start setup, I get the error "Unfortunately, the process android.process.media has stopped". During setup, I left the box "Back up your data to a Google Account. Restore previous backups to this device" checked during setup (with the other checkboxes on that setup screen unchecked). With those settings, I get the following behavior:
All of my apps start updating, until it tries to update Google Wallet, Maps and Calendar. I get a notification that those three apps require approval. Then Hangouts tries to update, but it notifies me that "Hangouts requires an update to Google Play Services". Sometimes, at this point, I'll get the android.process.media has stopped error again. Soon after the notification appears, the device will freeze for 1-2 seconds, and then will go back to the bouncing circles.
I'm guessing I need to completely reset the device to fix this issue. I found a guide on restoring a Nexus 7 back to the full factory image. Is that what I should try next, or are there easier things I should try next?